Here is a short file I put together regarding PRO-2004 repair. The printed circuit board problem referenced near the end of the text is the most common failure. I've repaired several PRO-2004's with this problem and it can take up to 40 hours to redo all the connections. In general it isn't worth the time to get a dead PRO-2004 going again. Since your unit does power up the 1 Ohm resistor is not the problem. You might consider selling the radio for parts either here in the classifieds forum or on Ebay.
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Repairing the PRO-2004 Scanner
The PRO-2004 radio scanner was made for Radio Shack by GRE, a Japanese electronics company. Introduced in late 1986, nearly 45,000 units were made by the time production ended in early 1988. An instant classic, the PRO-2004 was head and shoulders above any other scanner made when it was introduced. Even today it is one of the finest scanners available for conventional scanning.
One common problem with the PRO-2004 involves the memory backup battery circuit. Sometimes the low battery warning keeps activating even when a fresh battery is installed. Memory loss when the radio is unplugged is also a symptom of the same problem. The tips from Bob Parnass listed below are a good place to start to fix this problem.
Make sure the contacts on the battery connector are tight. The snaps widen and lose contact, and you can squeeze them gently with a pliers to tighten them.
Unplug the AC cord from power and remove the battery. After a few minutes, insert a good battery, plug the power cord back in the wall and perform a full reset according to the instruction manual.
Check the through-board solder joint at connector CN6, pin 3. Some owners have reported a bad joint there.
In addition to Bob's suggestions, be sure to check the wires from the battery connector to the printed circuit board for continuity. Use an Ohmmeter to be sure the connection is good.
If a new battery goes dead after just a few days, it is likely that the memory regulator has gone bad. The regulator is in a TO-92 package and looks like a small transistor with three leads. It is labeled IC9 on the printed circuit board. The part is made by Seiko, type S81250HG. The Radio Shack part number is 10889566 although it may no longer be available. Part number 628-812C50AY-G, available from Mouser Electronics, may be used as a substitute.
The PRO-2004 had serious quality problems with its printed circuit boards, especially during 1987 production. There are copper traces on the top and bottom of the boards. In places where it is necessary to connect top and bottom traces together, a hole was drilled and then plated with copper to make the connection. This plating was faulty and over time it cracks, breaking connection from top to bottom. The symptoms can range from corrupted memory to the squelch not working. Sometimes the problem comes and goes or may be sensitive to temperature. The only way to permanently fix the problem is to put wires in all the holes and solder the connections on top and bottom. Pop the tops off the shields and repair the connections inside too. This takes a huge amount of time and for this reason repair shops like G&G consider the PRO-2004 to be unrepairable.
Another problem involves the glue GRE used on the large power supply capacitors to secure them to the main PC board. It's often black in color and you should see it around the base of the capacitors. There is a corrosive element in the glue and given enough time it eats through the copper on the PC board. The glue can be scraped off the board with a toothpick (don't use a metal tool) and you can then assess the amount of damage due to corrosion. Repair any broken traces with small lengths of wire.